Strange problem about the huge surface temperature jump ?

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The outside of a construction is exposed to the exterior weather conditions and sun, you should expect its temperatures to be more dynamic than the inside surface of the construction. The less mass that the outside material of the construction has and the less conductive the adjacent material layer is, the quicker the temperatures of the outside surface will swing.

As an example think of concrete wall with rigid insulation and then a thin sheet of metal on the outside. It's been 50 overnight and now it is 80 and sunny. It isn't going to take long at all for the metal to heat up, but it will take a long time for the concrete to heat up, and even then, if the space is being conditioned, it is going to dampen any effect on the inside surface temperature.

This is really the goal of using insulation or thermal mass, and avoiding thermal bridging, in an envelope. To protect the inside of the building from the exterior conditions. If you are trying to lower the outside surface temperatures you could use surface finish with a high solar reflective index (SRI).